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Chpt 12 - 20 > Chpt 13 > Flashcards

Chpt 13 Flashcards

1
Q

What does ANOVA stand for

A

Analysis of Variance

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2
Q

what is the F- ratio

A

is the ratio of 2 variables

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3
Q

What does ANOVA allow us to do?

A

allows us to compare multiple pops and even subgroups of these pops
- how two groups interact with each other quantitatively

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4
Q

What question does ANOVA help us answer

A

do all 3 means come form a common population
- we are not asking if they were exactly equal. we are asking if each mean likely came from the larger overall population

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5
Q

What is the Null hypothesis for ANOVA

A

HO= M1 = M2 = M3

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6
Q

What is the problem with using pairwise comparison for 3 pop means

A

the type I error will compound with each t-test
95% confidence = (.95)(.95)(.95) = .857
so, a (or critical value) would be come 1 - .857 =
143

Type 1 error rate went from 5% (0.05) to 14.3%

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7
Q

what is partitioning

A

separating total variance into its component parts

- we do this by using ANOVA

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8
Q

What is the variability between the means

A

distance from overall mean

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9
Q

if the variability between the means (distance from overall mean) in the numerator is relatively Large compared to the variance within the samples the ratio will be

A

much larger than 1

  • the samples mostly likely do NOT come from a common pop
  • reject Ho that means are equal
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10
Q

What is the variability within the samples called

A

internal spread (the denominator)

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11
Q

If the F ratio is similar/similar what does this tell us

A
  • Fail to reject Ho

- means are fairly close to overall mean and/ or distributions overlap a bit

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12
Q

If the F ratio is Small /Large

A
  • Fail to reject Ho

- the means are very close to overall mean and/or distributions melt together

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13
Q

What is the formula for f ratio

A

B/W / W/in or Among / Around

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14
Q

variance b/w + Variance w/in (error variance) =

A

Total variance

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15
Q

Factor definition

A

independent variable (ie. assembly method)

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16
Q

What are the required assumptions for ANOVA

A
  1. normally distributed
  2. distributions must be independent
  3. the variance of the response variable (Qsquared) is the same for all pops
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17
Q

What are the steps to ANOVA

A
  1. Calculate sample mean for each pop
  2. calculate overall mean for all pops (add up all means / # of means)
  3. Estimate the variance (Xbar 1- Overall mean)squared / n-1
  4. compute the sum of squares b/w treatments
  5. computer mean squares b/w treatments
  6. calculate sum of squares due to error
  7. calculate the mean squares due to error
  8. Setup the ANova table
  9. calculate f-ratio and p-value
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18
Q

what is SSTR stand for

A

sum of squares b/w treatments

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19
Q

what is MSTR

A

mean square b/w treatments

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20
Q

what is SSE

A

sum of squares due to error

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21
Q

What is MSE

A

mean square due to error

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22
Q

what is the formula for SSTR

A

sum (# of sample)(pop1 mean - overall mean)sqaured (do for each set of pops)

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23
Q

What is the formula for MSTR

A

SSTR/k-1

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24
Q

What is the formula for SSE

A

(# of samples)(Variance of pop1) + (# of samples) (Variance of pop2) + (# of samples)(Variance of Pop3)

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25
Q

What is the formula for MSE

A

SSE/nr-k

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26
Q

What is the F-ratio formula

A

MSTR/MSE

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27
Q

what is k-1

A

3 pops - 1 = degrees of freedom

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28
Q

What is nr-k

A

total # of sample for all 3 pops (ex. each contain 5 samples than nr = 5x3
k = total pops (in this case 3)
so df = 15 - 3

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29
Q

What is Fishers LSD

A

remember ANOVA tells us if at least 2 of the groups are different from each other
- Fisher’s LSD tests 2 specific groups against each other

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30
Q

what does LSD stand for

A

LEast Significant Difference

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31
Q

What is the formula for Fisher’s LSD

A

t a/2 x square root of MSE (1/n1 + 1/n2)

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32
Q

what is t a/2

A

critical value using within degrees of freedom and alpha / 2

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33
Q

What do you compare LSD to

A

(xbari - xbarj)

  • reject if (xbar i - xbarj) is greater than or equal to LSD
  • do this for each group
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34
Q

LSD is used to determine

A

where the differences occur

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35
Q

what is the null hypothesis for LSD

A

HO: Mi = Mj

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36
Q

what is the test statistic for LSD

A

t = (xbar i - xbarj) / square root of MSE (1/ni + 1/nj)

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37
Q

What is the rejection rule for LSD - pvalue approach

A

reject HO if Pvalue is less than or equal to a (CV)

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38
Q

what is the rejection rule of LSD - cv approach

A

reject HO if t is less than or equal to - t a/2 or

t is greater than or equal to t a/2

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39
Q

what is the rejection rule of LSD - cv approach

A

reject HO if t is less than or equal to - t a/2 or

t is greater than or equal to t a/2

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40
Q

what is the degrees of freedom for LSD and t- distribution

A

t a/2 is based on a t-distribution with nT-k degrees of freedom

what is T???

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41
Q

LSD and Confidence intervals - if the confidence interval includes the value 0

A

we cannot reject Ho, that the pop means are equal

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42
Q

If the LSD confidence interval does not include the value 0

A

we can conclude there is a difference in pop means

- do not reject Ho

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43
Q

what is a Comparisonwise Type1 Error rate

A

indicate the level of significance associated with a single pairwise comparison a = 1-.95 = 0.05

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44
Q

What is a experimentwise type 1 error rate

A

Prob we will not make a type 1 error for all 3 tests

.95)(.95)(.95
- this gets larger the more groups you have

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45
Q

What is the experimentwise type 1 erorr rate denoted as

A

aEW

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46
Q

What is the experimentwise type 1 erorr rate denoted as

A

aEW

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47
Q

How do we control the overall experimentwise error rate

A

use Bonferroni Adjustment

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48
Q

What is the Bonferroni Adjustment

A

we use smaller comparisonwise error rate for each test

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49
Q

What is the formula for Bonferroni adj

A

aEW / C (C to test c pairwise comparisons)

ex.
a = 0.05 / 3 pops = 0.017

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50
Q

What are some other procedures we could use to control the overall experimentwise error rate

A
  1. Tukey’s procedure

2. Duncan’s multiple range test

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51
Q

When is randomized block design used

A

useful when the experimental units are homogenous

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52
Q

What do we use if exeperimental units are heterogenoeous

A

Blocking is often used to form homogenous groups

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53
Q

Problem with Randomized block design? (double check this is what it is referring to)

A

can arise whenever differences due to extraneous factors (ones not considered in the experiment) cause the MSE term to become too LARGE
- this can cause the f-value to be small, signaling no difference among treatment means when in fact a difference exists

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54
Q

HOw do you compute f-ratio for randomized block design

A

F = MSTR/MSE

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55
Q

In our example what would the workstation be

A

the factor of interest

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56
Q

in our example of randomized block design what would the controllers be

A

the blocks

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57
Q

what would the treatments be in a randomized block design

A

the pops

- 3 treatments (or pops) associated with workstation factor correspond to the 3 workstation alternatives

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58
Q

what would the treatments be in a randomized block design

A

the pops

- 3 treatments (or pops) associated with workstation factor correspond to the 3 workstation alternatives

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59
Q

What is the randomized aspect

A

is the random order in which the treatments (systems) are assigned to controllers

  • 6 controllers were selected at random and assigned to operate each of the systems
  • a follow up interview and a medical exam of each controlelr in the study provided a measure of stress for each controller on each system
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60
Q

What is SST = for randomzied block design

A

SST = SSTR + SSBL + SSE

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61
Q

What does k represent in randomized block design

A

the # of treatments

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62
Q

What does b represent in randomized block design

A

of blocks

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63
Q

What does nT represent in randomzied block design

A

total sample size (nT = kb)

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64
Q

What are the steps in randomized block design

A
  1. compute SST (total sum of squares)
  2. Compute SSTR (Sum of squares due to treatments)
  3. Compute SSBL Sum of Squares due to blocks
  4. Compute SSE (sum of squares due to error)
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65
Q

What is the formula in randomized block design for SSE

A

SSE = SST - SSTR - SSBL

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66
Q

What is the formula in randomized block design for SST

A

sum (Xbar - total block mean) squared

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67
Q

What is the formula in randomized block design for SSTR

A

(# in sample){sum (treatment mean - block mean)squared

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68
Q

What is the formula for SSBL

A

(# of pops) [(block mean - total block mean) squared]

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69
Q

What does SSBL mean

A

sum of squares due to blocks

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70
Q

What is SST

A

Total sum of squares

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71
Q

what is the degrees of freedom for SSTR

A

k- 1 ( # of pops - 1)

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72
Q

what is the degrees of freedom for SSBL in randomized block design

A

b-1 (# of blocks -1)

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73
Q

What is the degrees of freedom for SSE in randomized block design

A

(k-1)(b-1)

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74
Q

What is the degrees of freedom for SST in randomized block design

A

nT-1

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75
Q

read notes - i left some out

A

ready notes i left some out

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76
Q

Describe a factorial experiment

A

an exerimental design that allows simultaneous conclusions about 2 or more factors

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77
Q

Why use Factorial

A

used becasue the experimental conditons include all possible combinations of hte factors

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78
Q

Give an example of a Factorial Experiment

A

study involving (GMAT)
- scores range form 200 to 800
higher scores imply higher aptitude
- to impreove the GMAT scores, consider 3 prep programs
- each program has 3 treatments (the program they are in business, Engineering, Arts)
- second factor - whether a student’s undergrad affects the GMAT score (college)

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79
Q

What would be if we have 3 treatments (prep programs for GMAT) combinations in factorial design if we have 2 factors
factor 1 - the prep program
factor 2 - college attended

A

3 x 3 = 9 treatment combinations

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80
Q

What is replications

A

the sample size of 2 for each treatment combination indicates we have 2 replications

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81
Q

What is the formula in Block design for SST

A

sum (sample - overall mean)sqaured (for all samples)

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82
Q

What is the formula in block design for SSTR

A

of blocks [(treatment mean - overall treatment mean) sqaured) + (Treatment mean#2 - overall treatment mean) squared) + (treatment mean #3 - overall treatment mean) Squared)

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83
Q

ANOVA Table Deifnition

A

A table used to summarize the analysis of variance computations and res­ ults. It contains columns showing the source of variation, the sum of squares, the degrees of freedom, the mean square, and the F value(s)

84
Q

Blocking Definiton

A

The process of using the same or similar experimental units for all treat­ ments. The purpose of blocking is to remove a source of variation from the error term and hence provide a more powerful test for a difference in population or treatment means.

85
Q

Comparisonwise Type I error rate - Definition

A

The probability of a Type I error associated with a single pairwise comparison.

86
Q

Completely randomized design - Definition

A

An experimental design in which the treatments are randomly assigned to the experimental units.

87
Q

Experimental units - Definition

A

The objects of interest in the experiment

88
Q

Experimentwise Type I error rate - Definition

A

The probability of making a Type I error on at least one of several pairwise comparisons

89
Q

Factor - Definition

A

Another word for the independent variable of interest

90
Q

Factorial Experiment - Definition

A

An experimental design that allows simultaneous conclusions about two or more factors

91
Q

Interaction - Definition

A

The effect produced when the levels of one factor interact with the levels of another factor in influencing the response variable.

92
Q

Multiple comparison procedures - Definition

A

Statistical procedures that can be used to conduct statistical comparisons between pairs of population means

93
Q

Partitioning - Definition

A

The process of allocating the total sum of squares and degrees of freedom to the various components.

94
Q

Randomized block design - Definition

A

An experimental design employing blocking.

95
Q

Replications - Definition

A

The number of times each experimental condition is repeated in an experiment

96
Q

Response variable - Definition

A

Another word for the dependent variable of interest.

97
Q

Single-factor experiment - Definition

A

An experiment involving only one factor with k populations or treatments

98
Q

Treatments - Definition

A

Different levels of a factor

99
Q

If you have 300 treatments, total is 460 and you have 7 experimental units used for each of the 5 levels of the factor, what is the degrees of freedom for the 300 treatments?

A 
# of samples = 5 levels = 5  
n-1 = 5-1 = 4 
df = 4
100
Q

If you have 300 treatments, total is 460 and you have 7 experimental units used for each of the 5 levels of the factor, what is the sum of squares due to error and what is the degrees of freedom

A

460-300 = 160

df = total samples size - # of samples
= 7x5 =35 - 5 =30

101
Q

What is ANOVA interested in

A

the status of the populations that generate the data sets, and not in the data sets themselves

102
Q

What are some assumptions in ANOVA

A
  1. assume that the data in each data set have come form a single pop
  2. assume that all the pops have the same Q^2
103
Q

How do we interpret the variation in the data sets for ANOVA

A

we interpret the variation in each data set as being caused by small and random sources, collectively called error

104
Q

What is a question regarding ANOVA’s Errors

A

the question, then, is should we regard the variation in the values across the data sets as “error” or should they be attributed to some other source of variation that is not random (ie different pops)

105
Q

What is used as the benchmark in ANOVA

A

variation within the data sets (SSE)

106
Q

what is used to compare the benchmark to in ANOVA

A

variation between the data sets (SSTR) is compared

107
Q

In ANOVA if the between variation is much larger than the within variation, we may conclude what

A

that the data sets have in fact been generated form different populations

108
Q

If we assume that the pops have the same Q^2, we can pool these variations and obtain one measure of the

A

within variation (SSE) - this is the benchmark

109
Q

MSE is an extension of the concept of what

A

pooled variance

110
Q

in the case of two samples, the formula for MSE reduces to what

A

S^2p

111
Q

In the case of two samples, we can prove that the F ratio obtained form the ANOVA table equals the

A

square of the t value obtained from applying the two-sample t test
F=t^2

112
Q

The F test is a direct extension of the

A

t test for testing the equality of the population means of several populations, with the assumption that the populations are normally distributed with a common variance

113
Q

The total sum of squares and the total degrees of freedom are ________. so, any variation and any degrees of freedom left over from the total variation and total degrees of freedom unaccounted for by the “between” source goes to the ______ _______.

A

fixed

Within source

114
Q

What does the mean squares column in the ANOVA table give us

A

the relative effect of each source.

115
Q

If relatively speaking, the systematic effects are larger than the random effects, this results in a ________F value and the hypothesis of equal means must be -_______

A

Large F value

rejected

116
Q

What is the systematic effect

A

Between Source

117
Q

What is the nonsystematic or random effects called

A

within source

118
Q

What are the 3 assumptions required to use ANOVA

A
  1. for each population, the response variable is normally distributed
  2. The variance of the response variable (Q^2), is the same for all the populations
  3. THe observations must be independent
119
Q

If the sample sizes are equal, ANOVA is not sensitive to what

A

to the departure of the assumption of normally distributed populations

120
Q

If the sample sizes are equal, ANOVA is not sensitive to what

A

to the departure of the assumption of normally distributed populations

121
Q

If the means for the 3 pops are equal, we would expect what

A

the three sample means to be close together

122
Q

the closer the 3 sample means are to one another, the

A

weaker the evidence we have for the conclusion that the pop means differ

123
Q

If the variability among the sample means is “small”, it supports what

A

HO

124
Q

If the variability among the sample means is “Large” it supports what

A

Ha

125
Q

The between treatments estimate of Q^2 is based on the assumption that the

A

null hypothesis is true (Ho is true)

126
Q

Does the variation within each of the samples have an effect on the conclusion as well

A

yes

127
Q

When a simple random sample is selected from each pop, each of the sample variances provides what

A

un unbiased estimate of Q^2

128
Q

Why do we call Pooled or within-treatments estimate of Q^2

A

because each sample variance provides an estimate of Q^2 based only on the variation within each sample, the within-treatments estimate of Q^2 is not affected by whether the pop means are equal

129
Q

When the samples sizes are equal, the within-treatments estimate of Q^2 can be obtained by computing what

A

the average of the indivdiual sample variances

130
Q

Between-Treatments approach provides a good estimate of Q^2 only if what

A

the null hypothesis is true

131
Q

If the null hypothesis is false in ANOVA, the between0treatments appaorch does what

A

overestimates Q^2

132
Q

The within treatments approach provides what for the Q^2

A

provides a good estimate of Q^2 in whether the HO is true or the HA

133
Q

If the null hypothesis is true in ANONVA, the two estimates will be what

A

similar and their ratio will be close to 1

134
Q

When do we need to use multiple comparison procedures?

A

whenever we are performing a series of tests and are concerned with the overall level of significance attached to the whole experiment. When there are several tests, each at some level of significance a, although we still have control over the probability of TYpe I error for individual tests, we have no such control over the series of tests. Multiple comparison helps us out in this regard

135
Q

What is an example of a multiple comparison procedure for ANOVA

A

LSD?

136
Q

If the null hypothesis is true, MSTR and MST provide what

A

two independent and unbiased estimates of Q^2

137
Q

The between treatments approach in ANOVA provides what

A

a good estimate of Q^2 ONLY if HO is true
if the null hypothesis is true, then this estimate and the within treatments estimate will be similar and their ratio will be close to 1

138
Q

The within treatments approach in ANOVA provides what

A

a good estimate of Q^2 regardless if HO is true of not

139
Q

ANOVA is based on the development of two independent estimates of the common what

A

population variance of Q^2

140
Q

What are the two independent estimates of variance in ANOVA

A
  1. SSTR - B/w treatments

2. SSE - Within Treatments

141
Q

What are the two independent estimates of variance in ANOVA

A
  1. SSTR - B/w treatments

2. SSE - Within Treatments

142
Q

By comparing SSTR and SSE what can we determine

A

whether the population means are equal

143
Q

ANOVA is most used for how many pops

A

3 or more but can be used for two when testing the means of two pops are equal but doesn’t usually happen (use the x^2 test instead)

144
Q

How do you calculate the overall mean if the sample sizes are not all the same?

A

sum of all of the observations / the total # of observations

145
Q

If H0 is true, MSTR provides

A

an unbiased estimate of Q^2

146
Q

if the means of the k populations are not equal, MSTR is

A

not an unbiased estimate of Q^2

147
Q

When does MSTR over estimate Q^2

A

When HO is rejected

148
Q

What is MSE based on

A

based on the variation within each of the treatments; it is not influenced by whether the null hypothesis is true.

149
Q

Is MSE influenced by the null hypothesis HO?

A

no

150
Q

if the null hypothesis is false, the value of MSTR/MSE will be_______ be­cause MSTR _________Q^2.

A

inflated

Overestimates

151
Q

What is the test statistic in ANOVA

A

F = MSTR/MSE

152
Q

What can SST be Partitioned into

A

Two different sums of squares: SSTR and SSE

SSTR + SSE = SST

153
Q

If sample sizes are not equal, what must you do for LSD

A

you must calculate LSD for each one

154
Q

When the sample sizes are equal, what can you do with LSD

A

you only need to calculate one LSD

155
Q

What is fisher’s LSD used for

A

to determine where differences occur

156
Q

Why is LSD referred to as a protected or restricted LSD test

A

because it is employed only used if we first find a significant F value by using ANOVA

157
Q

In a One-way ANOVA (first part of chpt 13) we focus on test what

A

the effect of one independent variable

158
Q

What might a one way ANOVA not do

A

may not be able to detect differences in means if the differences are caused by another factor than the independent variable we are considering

159
Q

How can you overcome the limitation of one way ANOVA and testing the effect of the underlying factor is to do what

A

use the randomized block design

160
Q

What does the randomized block design allow us to test

A

the effect of the independent variable as well as the block effect

161
Q

What does a two way ANOVA allow us to do

A

test the effect of two or more independent variables and the interaction among these variables

162
Q

What type of ANOVA do we use if the exeperimental units are homogenous

A

completely randomized design

163
Q

What type of ANOVA do we use if the experimental units are heterogenous

A

blocking is often used to form homogenous groups

164
Q

What is the purpose of the block design

A

to control some of the extraneous sources of variation by removing such variation from the MSE term

165
Q

What does the randomized block design tend to provide

A

a better estimate of the true error variance and leads to a more powerful hypothesis test in terms of the ability to detect differences among treatment means

166
Q

Experimental studies in business often involve experimental units that are ____________; as a result, we should use _________________

A

highly heterogenous

randomized block design

167
Q

Blocking in experimental design is similar to what

A

Stratification in sampling

168
Q

what does nT represent

A

total sample size

169
Q

THe experimental design described in block design is a ________design. What does this mean

A

complete block design

the word complete indicates that each block is subject to all k treatments

That is all controllers (Blocks) were tested with all 3 systems (treatments)

170
Q

WHat is an incomplete block design

A

experimental designs in which some but not all treatments are applied to each block - not in this text

171
Q

what is important to note about the F tests in the block design

A

we have an F value to test for treatment effects but not for blocks
blocking was used to remove variation from the MSE term

could use MSB/MSE and use the static to test for significance of the blocks

172
Q

The error degrees of freedom are _______ for a randomized block design than for a completely randomized design because _______

A

are less

b-1 degrees of freedom are lost for the b blocks

173
Q

if n is small, the potential effects due to blocks can be

A

masked because the loss of error degrees of freedom; for large n, the effects are minimized

174
Q

if n is small, the potential effects due to blocks can be

A

masked because the loss of error degrees of freedom; for large n, the effects are minimized

175
Q

If we want to draw conclusions about more than one variable or factor what can we use

A

factorial experiment

176
Q

what is factorial experiment

A

an experimental design that allows simultaneous conclusions about two or more factors

177
Q

why do we use the term factorial in factorial experiment

A

because the experimental conditions include all possible combinations of the factors

178
Q

what does interaction in factorial design mean

A

refers to a new effect that we can now study because we used a factorial experiment

179
Q

If the interaction effect has a significant impact on what we studying (ie GMAT), we can conclude what

A

that the effect of the type of preparation program depends on the under grad college

180
Q

in two-factor we do an Mean square for

A

Factor A : MSA = SSA/ a-1

Factor B: MSB = SSB/ b-1

Interaction: MSAB = SSAB / (a-1)(b-1)

Error: MSE = SSE /ab(r-1)

181
Q

In two-factor we calcualte F for

A

Factor A: MSA/MSE

Factor B: MSB/MSE

Interaction: MSAB/MSE

182
Q

define factor

A

the independent variable of interest

183
Q

define treatments

A

different levels of a factor

184
Q

single-factor experiment - define

A

an experiment involving only one factor with k populations or treatments

185
Q

define response variable

A

another word for the dependent variable of interest

186
Q

define experimental units

A

the objects of interest in the experiment

187
Q

define completely randomized design

A

an experimental design in which the treatments are randomly assigned to the experimental units

188
Q

define ANOVA table

A

a table used to summarize the analysis of varaiance compuations and results

189
Q

Define Partitioning

A

the process of allocating the total sum of squares and degrees of freedom to the various components

190
Q

define Multiple comparison procedures

A

statistical procedures that can be sued to conduct satistical comparisons between pairs of population means

191
Q

define comparisonwise TYpe 1 error rate

A

the provability of a type 1 error associated with single pairwise comparison

192
Q

define Experimental TYpe I error rate

A

the probability of making a type 1 error on at least one of several pairwise comparisons

193
Q

define blocking

A

the process of using the same or similar experimental units for all treatments.

194
Q

What is the purpose of blocking

A

is to remove a source of variation from the error term and hence provide a more powerful test for a difference in population or treatment means

195
Q

Define randomized block design

A

an experimental design employing blocking

196
Q

define factorial experiment

A

an experiment design that allows simultaneous conclusions about two or more factors

197
Q

Define replications

A

the number of times each experimental condition is repeated in an experiment.

198
Q

Define interaction

A

the effect produced when the levels of one factor interact with the levels of an other factor in influencing the response variable

199
Q

What does the two -way anova have the added advantage of

A

allowing us to study the interaction effect between the variables

200
Q

With the TWO way ANOVA, when interpreting the results, it is a good idea to focus on what

A

the interaction effect first.

201
Q

in a two way ANOVA, if the interaction effect proves to be significant, what do you do

A

a further detailed analysis can be applied to this aspect

202
Q

In a two way ANOVA, if the interaction effect provides to insignificant, what can you do

A

focus can be directed on the main effects

203
Q

What is a factor in ANOVA

A

the I.V.

204
Q

THe factor is also a

A

variable of interest

205
Q

A treatment is

A

different levels of a factor

Chpt 12 - 20 (10 decks)

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